The Lip-lip equality is stable under blow-up

TL;DR

This paper proves that the Lip-lip equality remains stable under blow-up processes in differentiability spaces, ensuring that tangent spaces preserve key analytic properties and enabling characterization of the p-weak gradient in iterated blow-ups.

Contribution

It demonstrates the stability of the Lip-lip equality under blow-ups and characterizes the p-weak gradient in iterated differentiability space blow-ups.

Findings

Tangents of differentiability spaces are themselves differentiability spaces.

The Lip-lip equality passes to tangent spaces.

Characterization of the p-weak gradient on iterated blow-ups.

Abstract

We show that at generic points blow-ups/tangents of differentiability spaces are still differentiability spaces; this implies that an analytic condition introduced by Keith as an inequality (and later proved to actually be an equality) passes to tangents. As an application, we characterize the $p$-weak gradient on iterated blow-ups of differentiability spaces.